Lesson 4
Understanding Decay
Problem 1
A new bicycle sells for $300. It is on sale for \(\frac{1}{4}\) off the regular price. Select all the expressions that represent the sale price of the bicycle in dollars.
\(300 \boldcdot \frac{1}{4}\)
\(300 \boldcdot \frac{3}{4}\)
\(300 \boldcdot \left(1 - \frac{1}{4}\right)\)
\(300 - \frac{1}{4}\)
\(300 - \frac{1}{4} \boldcdot 300\)
Solution
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Problem 2
A computer costs $800. It loses \(\frac{1}{4}\) of its value every year after it is purchased.
- Complete the table to show the value of the computer at the listed times.
- Write an equation representing the value, \(v\), of the computer, \(t\) years after it is purchased.
- Use your equation to find \(v\) when \(t\) is 5. What does this value of \(v\) mean?
time (years) |
value of computer (dollars) |
---|---|
0 | |
1 | |
2 | |
3 | |
\(t\) |
Solution
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Problem 3
A piece of paper is folded into thirds multiple times. The area, \(A\), of the piece of paper in square inches, after \(n\) folds, is \(A = 90 \boldcdot \left(\frac{1}{3}\right)^n\).
- What is the value of \(A\) when \(n = 0\)? What does this mean in the situation?
- How many folds are needed before the area is less than 1 square inch?
- The area of another piece of paper in square inches, after \(n\) folds, is given by \(B = 100 \boldcdot \left(\frac{1}{2}\right)^n\). What do the numbers 100 and \(\frac{1}{2}\) mean in this situation?
Solution
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Problem 4
At the beginning of April, a colony of ants has a population of 5,000.
- The colony decreases by \(\frac{1}{5}\) during April. Write an expression for the ant population at the end of April.
- During May, the colony decreases again by \(\frac{1}{5}\) of its size. Write an expression for the ant population at the end of May.
- The colony continues to decrease by \(\frac{1}{5}\) of its size each month. Write an expression for the ant population after 6 months.
Solution
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Problem 5
Lin has 13 mystery novels. Each month, she gets 2 more. Select all expressions that represent the total number of Lin's mystery novels after 3 months.
13 + 2 + 2+ 2
\(13 \boldcdot 2 \boldcdot 2 \boldcdot 2\)
\(13 \boldcdot 8\)
13 + 6
19
Solution
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(From Unit 5, Lesson 2.)Problem 6
An odometer is the part of a car's dashboard that shows the number of miles a car has traveled in its lifetime. Before a road trip, a car odometer reads 15,000 miles. During the trip, the car travels 65 miles per hour.
- Complete the table.
- What do you notice about the differences of the odometer readings each hour?
- If the odometer reads \(n\) miles at a particular hour, what will it read one hour later?
duration of trip (hours) |
odometer reading (miles) |
---|---|
0 | |
1 | |
2 | |
3 | |
4 | |
5 |
Solution
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(From Unit 5, Lesson 2.)Problem 7
A group of students is collecting 16 oz and 28 oz jars of peanut butter to donate to a food bank. At the end of the collection period, they donated 1,876 oz of peanut butter and a total of 82 jars of peanut butter to the food bank.
- Write a system of equations that represents the constraints in this situation. Be sure to specify the variables that you use.
- How many 16 oz jars and how many 28 oz jars of peanut butter were donated to the food bank? Explain or show how you know.
Solution
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(From Unit 2, Lesson 12.)Problem 8
A function multiplies its input by \(\frac 3 4\) then adds 7 to get its output. Use function notation to represent this function.
Solution
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(From Unit 4, Lesson 4.)Problem 9
A function is defined by the equation \(f(x) = 2x - 5\).
- What is \(f(0)\)?
- What is \(f(\frac 1 2)\)?
- What is \(f(100)\)?
- What is \(x\) when \(f(x) = 9\)?
Solution
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(From Unit 4, Lesson 4.)